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Class 10th ICSE OBJECTIVE TYPE QUESTIONS ON LINEAR INEQUATIONS
What are Linear Inequations?
10th ICS10th MCQ On Linear Inequations Linear inequalities are the expressions where any two values are compared by the inequality symbols. These are the equations that contain all mathematical symbols except equal to (=). The values can be numerical or algebraic or a combination of both. The inequality symbols are < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to). The symbols < and > shows the strict inequalities and the symbols ≤ and ≥ represents the slack inequalities.
Example:
x < 3, x ≥ 5, y ≤ 8, p > 10, m ≠ 1.
Linear Inequalities Rules
Solving a Linear Inequation Algebraically: To solve a given linear inequation means to find the value or values of the variable used in it. Thus, to solve the inequation 3x+5>8means to find the variable x and to solve the inequation 8–5y<3means to find the variable y and so on.
The following working rules must be adopted for solving a given linear inequation:
Rule 1: On transferring a positive term from one side of an inequation to its other side, the sign of the term becomes negative.
Rule 2: On transferring a negative term from one side of an inequation to its other side, the sign of the term becomes positive.
Rule 3: If each term of an inequation is multiplied or divided by the same positive number, the sign of inequality remains the same.
That is if pp is positive and p=0.p=0.
- x<y⇒px<py
2. x>y⇒px>py
3. x≤y⇒px≤py
4. x≥y⇒px≥py
Rule 4: If each term of an inequation is multiplied or divided by the same negative number, the sign of inequality reverses. That is if p is negative.
- x<y⇒px>py
- x≥y⇒px≤py and so on
Rule 5: If the sign of each term on both sides of an inequation is changed, the sign of inequality gets reversed.
Rule 6: If both the sides of an inequation are positive or both are negative, then on taking their reciprocals, the sign of inequality reverses.
That is if xx and yy both are either positive or both are negative, then:
1. x>y⇔1/x<1/y
2. x≤y⇔1/x≥1/y and so on.
Replacement Set and Solution Set
Replacement Set: The set from which values of the variable involved in the inequality are chosen is called the replacement set. Generally, we use either N (set of natural numbers; 1,2,3,4,….1,2,3,4,…. ) or W (set of whole numbers; 0,1,2,3,….0,1,2,3,….) or I (set of integers; …–3,–2,–1,0,1,2,3,….…–3,–2,–1,0,1,2,3,….) or R (real number) as the replacement set. (OBJECRTIVE TYPE QUESTIONS ON LINEAR INEQUATIONS )
Solution Set: The solution to an inequation is a number that, when substituted for the variable, makes the inequation true. The set of all solutions of the given inequation is called the solution set of the inequation. The solution set is the subset of the replacement set.
Method to Solve a Linear Inequation in One Variable
There are the following steps used to solve the linear inequation in one variable.
- Remove the fractions (or decimals) by multiplying both sides by an appropriate factor.
2. Put all variable terms on one side and all constants on the other side.
3. Make the coefficient of the variable 1.
4. Choose the solution set from the replacement set.
Representation of Solution Set of Linear Inequation in One Variable on Number Line
Use the following rules to represent the solution of a linear inequation in one variable on the number line
1. If the inequation involves ≤ or ≥ drawing a filled circle or dark circle (.) on the number line, then the number corresponding to the filled circle or dark circle is included in the solution set.
For example: If 2 is also included, i.e., x≤2,, then the circle will be darkened.
2. If the inequation involves > or <,, then draw an unfilled circle or blank circle (o) on the number, the line shows that the number corresponding to the unfilled circle or blank circle is excluded from the solution set.
For example x<2 and x∈R
Important questions for OBJECTIVE TYPE QUESTIONS ON LINEAR INEQUATIONS
Answer Key For Linear Inequations
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10th ICSE OBJ10th MCQ On Linear Inequations